Finally, the conceptual understanding of renormalization has shown (see [1]})
that there is a highly noncommutative symmetry group, the Cosmic Galois group, which embodies the ambiguities in the renormalization process inherent to the physics computations of interacting Quantum Field Theory. How this symmetry group combines with the above impact of physics on geometry, and how renormalization fits with the spectral action principle are deep open questions of which only the first few steps have recently been taken [2]}, [3]}. These questions are witnesses of the amazing role of noncommutativity in physics.
